Preheat the oven to 325 degrees.
In a very large heavy, deep ovenproof casserole, heat 2 tablespoons oil on medium-high heat. Dry the brisket well. Season the brisket with salt and pepper on both sides and brown the brisket, about 4 minutes per side, until nicely browned. (This will assure the sauce is a deep brown rich color and flavor.) Remove the brisket to a large dish and reserve.
Add the remaining oil to the pan on medium heat and sauté the onions and leeks until a rich, golden brown color, scraping up the brown bits from the bottom of the pan as the onions cook, about 5 to 20 minutes. Add the carrots and garlic and cook another 2 minutes. Add the beer and combine with the onions. Season with salt and pepper.
Return the brisket on top of the onion mixture in the casserole and glaze the brisket evenly with the sun-dried tomato paste, by using a spatula to evenly spread over the top of the brisket. Cover and transfer to the oven to braise for 11/2 hours.
Remove the brisket to a carving board and slice the meat against the grain, placing the meat back into the casserole in overlapping slices.
Return the brisket to the oven and continue cooking another 11/2 to 2 hours or until very tender when pierced with a fork. To serve, place the brisket slices in a serving dish with edges overlapping to catch the sauce. Spoon over the sauce, garnish with parsley and serve immediately.